Mathematical Topic: Exponentials and Logarithms

Internal combustion engines

Post date 30 September 2021

A country’s government wants to reduce the number of cars using internal combustion engines by encouraging the purchase of electric vehicles.

The total number of cars using internal combustion engines in a country at time t can be modelled by the equation

\[N\left( t \right) = \left( 3.5 \times 10^{7} + E \right)e^{- \frac{1}{10}t} – E\]

Where N(t) is the number of cars using internal combustion engines at time t years after the program was introduced and E is the number of electric vehicles on the roads before the government initiative.

Using the equation for this model,

a) Explain the significance of the number 3.5 x 10⁷.

[1 mark]

b) Explain why the initial number of cars using internal combustion engines is independent of the initial number of electric vehicles.

[2 marks]

c) Calculate the number of cars using internal combustion engines after 10 years if the initial number of electric vehicles is 2 x 10⁶.

[2 marks]

d) Analyse the suitability of this model by evaluating it for large values of t.

[2 marks]

Shrinking Species

Post date 30 September 2021

Since 1800, the number of amphibian species, N, has been decreasing over time, t.

A simple model shows that the rate of decrease of the number of species is proportional to the remaining number of species.

Given that the initial number of amphibian species is N₀, and t is the number of years since 1800,

a) Show that \(N = N_{0}e^{- kt}\)

[4 marks]

In 2000 the number of amphibian species is 0.9N₀.

b) Find the exact value of k.

[3 marks]

c) Using the model, in what year will 20% of amphibian species be
extinct?

[3 marks]

The Swelling Sahara

Post date 30 September 2021

Human-induced global warming is causing deserts such as the Sahara to increase in surface area.

In 1950 the area of the Sahara Desert was 9,200,000 km², whereas in 2000 the area of the Sahara Desert had increased to 9,930,000 km² due to human-induced global warming.

A model could be used to relate the surface area of the Sahara Desert, S km², to the time, t, in years since 1950.

a) By first forming an exponential model for the surface area of the Sahara Desert relating S and t, show that the increase in the surface area of the Sahara Desert is approximately 0.15% per year.

[5 marks]

b) Use the model formed in (a) to estimate the size of the Sahara Desert in 2050. Give your answer in km² to 3 s.f.

[1 mark]

A Shrinking Rainforest

Post date 30 September 2021

In a simple model, the surface area, S km², of a shrinking rainforest depends on the time, t, in years since 1980.

The following information is available for rainforest A.

its surface area in 1980 was 300,000 km²
its surface area in 1981 was 294,000 km²

a) Use an exponential model to form, for rainforest A, a possible equation linking S with t.

[4 marks]

The surface area of rainforest A is monitored over a 30-year period. Its surface area after 30 years is 150,000 km².

b) Evaluate the reliability of your model in light of this information.

[2 marks]

The following information is known about rainforest B.

it had the same surface area, in 1980, as rainforest A
it is harder to access by road, so the rate of deforestation is less than rainforest B and its surface area decreases more slowly than that of rainforest A

c) Explain how you would adapt the equation found in (a) so that it could be used to model the surface area of rainforest B.

[1 mark]

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